Radar System Engineering

406 R-F COMPONENTS [SEC. 114
d forms the properties are completely calculable from Maxwell’s
equations.
In cavities of the type used in magnetrons, the inductance and
,:apacity are fairly well separated and approximate numerical values can
be calculated though they cannot be measured independently. For
something like a +,mple cylindrical cavity, however, inductance and
capacity are blended and cannot even be calculated unambiguously.
The significant quantity is the resonant frequency or wavelength.
The second property of cavities that is important in microwave work
is their Q. As in lumped-constant circuits, the value of Q is a measure
of the sharpness of the resonance, and is determined by the dissipative
elements loading the resonant circuit. If ~0is the resonant frequency and
~, and ~2 are the “ half-power points,” —that is, the two frequencies, one
above jo and one below jo, at which the voltage
is 0.707 as great as it is at resonance—then
(or current) in the cavity
(5)
An equivalent, but somewhat more general, formula for Q involves
the amount of electromagnetic energy stored in the oscillating field within
the cavity, and the rate at which energy is dissipated in the walls or in
any other way. If we denote the total stored energy by W, and the
energy dissipated during one r-f cycle by W, Q is given by
Q=_.
27rw
w
(6)
If w includes only the dissipation within the cavity itself, due to the
resistance of the walls and to dielectric losses in insulators within the
cavity, etc., the Q defined above is called the unloaded Q, usually written
Qo. If w includes, in addition, energy dissipated in external circuits
coupled to the cavity, we obtain instead the loaded Q, or QL, which of
course can never exceed Qo.
In a simple cylindrical cavity made of copper and resonant at 3000
Me/see in its lowest mode, the unloaded Q is about 15,000. Generally,
the Q of a cavity loaded only by the resistance of its walls depends on the
ratio of the volume of the cavity to the product of the internal surface
area and the skin depth. For cavities of similar shape, the resonant frequency
jo is inversely proportional to a linear dimension of the cavity;
the skin depth varies as l/fro. It follows that Qo, for cavities of similar
shape, varies as l/@o. On the other hand, the Q values that can be
attained at microwave frequencies, typified by the example just given,
are much higher than can bc realized with coil and condenser combina-